NAG FL Interface
g02aef (corrmat_​nearest_​kfactor)

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1 Purpose

g02aef computes the factor loading matrix associated with the nearest correlation matrix with k-factor structure, in the Frobenius norm, to a given square, input matrix.

2 Specification

Fortran Interface
Subroutine g02aef ( g, ldg, n, k, errtol, maxit, x, ldx, iter, feval, nrmpgd, ifail)
Integer, Intent (In) :: ldg, n, k, maxit, ldx
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: iter, feval
Real (Kind=nag_wp), Intent (In) :: errtol
Real (Kind=nag_wp), Intent (Inout) :: g(ldg,*), x(ldx,*)
Real (Kind=nag_wp), Intent (Out) :: nrmpgd
C Header Interface
#include <nag.h>
void  g02aef_ (double g[], const Integer *ldg, const Integer *n, const Integer *k, const double *errtol, const Integer *maxit, double x[], const Integer *ldx, Integer *iter, Integer *feval, double *nrmpgd, Integer *ifail)
The routine may be called by the names g02aef or nagf_correg_corrmat_nearest_kfactor.

3 Description

A correlation matrix C with k-factor structure may be characterised as a real square matrix that is symmetric, has a unit diagonal, is positive semidefinite and can be written as C=XXT+diag(I-XXT), where I is the identity matrix and X has n rows and k columns. X is often referred to as the factor loading matrix.
g02aef applies a spectral projected gradient method to the modified problem minG-XXT+diag(XXT-I)F such that xiT21, for i=1,2,,n, where xi is the ith row of the factor loading matrix, X, which gives us the solution.

4 References

Birgin E G, Martínez J M and Raydan M (2001) Algorithm 813: SPG–software for convex-constrained optimization ACM Trans. Math. Software 27 340–349
Borsdorf R, Higham N J and Raydan M (2010) Computing a nearest correlation matrix with factor structure SIAM J. Matrix Anal. Appl. 31(5) 2603–2622

5 Arguments

1: g(ldg,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array g must be at least n.
On entry: G, the initial matrix.
On exit: a symmetric matrix 12(G+GT) with the diagonal elements set to unity.
2: ldg Integer Input
On entry: the first dimension of the array g as declared in the (sub)program from which g02aef is called.
Constraint: ldgn.
3: n Integer Input
On entry: n, the order of the matrix G.
Constraint: n>0.
4: k Integer Input
On entry: k, the number of factors and columns of X.
Constraint: 0<kn.
5: errtol Real (Kind=nag_wp) Input
On entry: the termination tolerance for the projected gradient norm. See references for further details. If errtol0.0, 0.01 is used. This is often a suitable default value.
6: maxit Integer Input
On entry: specifies the maximum number of iterations in the spectral projected gradient method.
If maxit0, 40000 is used.
7: x(ldx,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array x must be at least k.
On exit: contains the matrix X.
8: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g02aef is called.
Constraint: ldxn.
9: iter Integer Output
On exit: the number of steps taken in the spectral projected gradient method.
10: feval Integer Output
On exit: the number of evaluations of G-XXT+diag(XXT-I)F.
11: nrmpgd Real (Kind=nag_wp) Output
On exit: the norm of the projected gradient at the final iteration.
12: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, k=value and n=value.
Constraint: 0<kn.
On entry, ldg=value and n=value.
Constraint: ldgn.
On entry, ldx=value and n=value.
Constraint: ldxn.
On entry, n=value.
Constraint: n>0.
Spectral gradient method fails to converge in value iterations.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The returned accuracy is controlled by errtol and limited by machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g02aef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02aef makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

Arrays are internally allocated by g02aef. The total size of these arrays is n×n+4×n×k+(nb+3)×n+n+50 real elements and 6×n integer elements. Here nb is the block size required for optimal performance by f08fef and f08fgf which are called internally. All allocated memory is freed before return of g02aef.
See g03caf for constructing the factor loading matrix from a known correlation matrix.

10 Example

This example finds the nearest correlation matrix with k=2 factor structure to:
G = ( 2 −1 0 0 −1 2 −1 0 0 −1 2 −1 0 0 −1 2 )  

10.1 Program Text

Program Text (g02aefe.f90)

10.2 Program Data

Program Data (g02aefe.d)

10.3 Program Results

Program Results (g02aefe.r)